Nested open sets?

1. Mar 18, 2012

cragar

1. The problem statement, all variables and given/known data
Give an example of an infinite collection of nested open sets.
$o_1 \supseteq o_2 \supseteq o_3 \supseteq o_4 .....$
Whose intersection $\bigcap_{n=1}^{ \infty} O_n$ is
closed and non empty.
2. Relevant equations
A set $O \subseteq \mathbb{R}$ is open if for all points, $a \in O$
there exists an $\epsilon$ neighborhood $V_{\epsilon}(a) \subseteq O$
3. The attempt at a solution
It seems like if we started with the open interval (0,1) and then took a smaller interval that was nested inside the original interval, and then just kept doing this until we enclosed one point in the interval.

2. Mar 18, 2012

micromass

Staff Emeritus
Yes, that's the idea.

Can you make this rigorous??

3. Mar 18, 2012

cragar

Can I just take the middle one-third of the set.
so after n operations i will have $\frac{1}{3^n}$

4. Mar 18, 2012

jgens

I cannot really tell where you are going with that last response. It might be a little easier if you consider intervals of the form $(-\frac{1}{n},\frac{1}{n})$.

5. Mar 18, 2012

micromass

Staff Emeritus
What do you mean?? 1/3^n is just a number.

Just find sets that get smaller and smaller each time.

6. Mar 18, 2012

cragar

ok so like jgens said use $( \frac{-1}{n} , \frac{1}{n})$
And then eventually after n goes to infinity I will have 0 as my enclosed point.
so if I make an $\epsilon$ radius around 0 i will contain points inside of O the original set. Would the set zero it self be closed be cause if we make
an $\epsilon$ radius around 0 it wont contain elements that are in the set zero itself.

7. Mar 18, 2012

HallsofIvy

Staff Emeritus
Or, if you really want $1/3^n$, you could use $O_n= (-1/3^n, 1/3^n)$.

8. Mar 18, 2012

cragar

ok, thanks everyone for the help